3 Systems of differential equations In Sections 3.8-3.10 we will reduce the problern of finding all solutions of (1) to the much simpler algebraic problern of solving simultaneous linear equations of the form allxl + a12X2 + + alnxn = bl a21X1 +a22x2+ +a2nxn=b2 Therefore, we will now digress to study the theory of simultaneous linear equations. Here too we will see the role played by linear algebra. EXERCISES In each of Exercises 1-4 find a basis for the set of solutions of the given differential equation. 1. x= ( _ ~ _ ~ )x (Hint: Find a second-order differential equation satis- fied by x 1(t).) 2. x=G -~ r)x (Hint: Find a third-order differential equation satis- fied by x 1(t).) 3. nx x=(i 4. x=O 011 00)1 X For each of the differential equations 5-9 determine whether the given solutions are a basis for the set of all solutions. 6. x=( -1 -3 7. x= ( s. x= ( -5 296 ~ 1 -1 -2 0 -2 -~)x; -1 3.5 The theory of determinants -31] , =~2) x; x1(t)= [:~31 10. Determine the solutions q, 1,q,2, ,q,n (see proof of Theorem 5) for the system of differential equations in (a) Problem 5; (b) Problem 6; (c) Problem 7. 11. Let V be the vector space of all continuous functions on (- oo, oo) to Rn (the values of x(t) lie in Rn). Let x 1,x2 , ,xn be functions in V. (a) Show that x 1(t0), ,xn(t0) linearly independent vectors in Rn for some t0 implies x 1,x2 , .. ,xn are linearly independent functions in V. (b) Is it true that x 1(t0), ,xn(t0) linearly dependent in Rn for some t0 implies x 1, x2, , xn are linearly dependent functions in V? Justify your ans wer. IZ. Let u be a vector in Rn(u~O). (a) Is x(t)= tu a solution of a linear homogeneous differential equation i=Ax? (b) Is x(t)=e;vu? (c) Is x(t)=(e 1 -e- 1)u? (d) Is x(t)=(e 1 +e- 1)u? (e) Is x(t)=(eA 1 +eA21)u? (f) For what functions q>(t) can x(t)=q,(t)u be a solution of some i=Ax? 3.5 The theory of determinants In this section we will study simultaneaus equations of the form + a 12x 2 + ... + a 1nxn = b 1 a21X 1 + a22X2 + + a2nxn = b2 a 11 x 1 an lXI (I) + an2x2 + ... + annxn = bn. Our goal is to determine a necessary and sufficient condition for the system of equations (I) to have a unique solution x 1,x2, ,xn. To gain some insight into this problem, we begin with the simplest case n = 2. If we multiply the first equation a 11 x 1 + a 12 x 2 = b 1 by aw the second equation a 21 x 1 + a22 x 2 = b 2 by a 11 , and then subtract the former from the latter, we obtain that (a 11 a 22 - a 12 a21 )x2 = a 11 b 2 - a 21 b 1. Similarly, if we multiply the first equation by a 22 , the second equation by a12, and then subtract the latter from the former, we obtain that ( aua22- a,2a2,)x, = a22bl- a,2b2. Consequently, the system of equations (I) has a unique solution 297 3 Systems of differential equations 7. Suppose that AD =DA for all matrices A. Prove that D is a multiple of the identity matrix. 8. Prove that AxadjA=detAxl. In each of Problems 9-14, find the inverse, if it exists, of the given matrix. 9. ( -~ ~ 4 11. ( 13. 1 ;) -1 10. (co~9 ~ u sin9 -i 1 -D i 12. (l 1 1f) -shn9) cos9 0 2 -1-~) -1~ -1 l) 1 3 1 l+i 0 14. ( - 15. Let 0 Show that if detA""'O. 16. Show that (AB)- 1 =B- 1A- 1 if detAXdetB""'O. In each of Problems 17-20 show that x = 0 is the unique solution of the given system of equations. 17. x 1 - x 2 - x 3 =0 3x 1 - x 2 +2x3=0 2x 1 +2x2 +3x 3=0 18. x 1 +2x2+4x3=0 x 2 + x 3 =0 x 1 + x 2 + x 3 =0 19. =0 x1+ 2x 2 - x3 2x 1+ 3x 2+ x3- x 4 =0 +2x 3+2x4 =0 -x1 3x 1x2+ x 3+3x4 =0 20. x 1 + 2x 2- x 3+3x4 =0 2x 1 + 3x 2 - x 4 =0 -x 1 + x 2+2x 3+ x 4 =0 -x2+2x 3+3x4 =0 3.7 Lineartransformations In the previous section we approached the problern of solving the equation Ax=b, an [ A= : (1) an1 by asking whether the matrix A - 1 exists. This approach led us to the conclusion that Equation (1) has a unique solution x=A- 1b if detA~O. In 320 3 Systems of differential equations 16. Let tt be a linear transformation taking Rn~Rn. Show that lt(O)=O. 17. Let ~(8) be the linear transformation which rotates each point in the plane by an angle 8 in the counterclockwise direction. Show that -sinB)(x ) cosB X2 18. Let ~ and ~ 2 be the linear transformations which rotate each point in the plane by angles n. and 82 respectively. Then the linear transformation ~ = ~ o ~ rotates each point in the plane by an angle 8 1 + 82 (in the counterclockwise direction). Using Exercise (15), show that -sin(8 1 +82 ) )=(cos8 1 sin8 1 cos(8 1 +82 ) 1 2) ( cos(8 +8 sin(8 1 +82 ) -sin8 1 )(cos82 cos8 1 sin82 Thus, derive the trigonometric identities sin(8 1 + 82 ) = sin8 1 cos82 + cos8 1sin82 cos( 81 + 82 ) = cos8 1 cos82 - sin8 1 sin82 19. Let lt(x 1, x 2 ) = ( ;; ~ ;~ ) (a) Verify that tt is linear. (b) Show that every point (x.,x2) on the unit circle xr+ xi= 1 goes into a point on the circle, xr + xi = 2. 20. Let V be the space of all polynomials p(t) of degree less than or equal to 3 and Iet (Dp)(t)=dp(t)/dt. (a) Show that D is a linear transformation taking V into V. (b) Show that D does not possess an inverse. 21. Let V be the space of all continuous functionsf(t),- oo < t < oo and Iet (Kf)(t) = Jo' f(s)ds. (a) Show that K is a linear transformation taking V into V. (b) Show that (DK)j= f where Dj= f'. (c) Let f(t) be differentiable. Show that [(KD)f](t)= f(t)- j(O). 22. A linear transformation tt is said tobe 1-1 if lt(x)*lt(y) whenever X*Y In other words, no two vectors go into the same vector under lt. Show that (t is 1-l if, and only if, lt(x)= 0 implies that x=O. 23. A linear transformation tt is said to be onto if the equation lt(x) = y has at least one solution for every y in Rn. Prove that (t is onto if, and only if, (t is 1-l. Hint: Showfirst that tt is onto if, and only if, the vectors lt(e 1), ... ,lt(en) are linearly independent. Then, use Lemma 1 to show that we can find a nonzero solution of the equation lt(x) =0 if lt(e 1), ... , lt(en) are linearly dependent. Finally, show that lt(e 1), ... , lt(en) are linearly dependent if the equation lt(x)=O has a nonzero solution. 332